Equitable Coloring of Sparse Planar Graphs

Abstract

A proper vertex coloring of a graph G is equitable if the sizes of color classes differ by at most one. The equitable chromatic threshold $\chi_{eq}^*(G)$ of G is the smallest integer m such that G is equitably n-colorable for all $n\geq m$. We show that for planar graphs G with minimum degree at least two, $\chi_{eq}^*(G)\leq4$ if the girth of G is at least 10, and $\chi_{eq}^*(G)\leq3$ if the girth of G is at least 14.